the chances are crazy low. iirc there was speculation that the gold 5 star on story maps has a lower drop rate than the limited drop on the same map. idk how true that is since over the course of trying to collect each chapters limited doll, ive collected all the other dolls in the pool as well.

i think the most runs i have on a boss, while going for the limited doll or the 5 star instead, is like over 300? but i also will send an auto battle to the other boss map, either normal or emergency, on a regular basis just to get a few more chances to pick up the dolls. i actually got the limited doll for one of the earlier maps from an auto battle but that was it, its never netted me any other ones.

its doable. also using production is a good idea. like running rifle recipe for your daily crafts. pretty low impact to your resources and over a long enough time will add up to a lot of chances at the doll you want. at the bare minimum it will get you other dolls out of the pool so you have options.

good luck.